A Combinatorial Analysis of Tree-Like Sentences

A sentence over a finite alphabet A , is a finite sequence of non-empty words over A . More generally, we define a graphical sentence over A by attaching a non-empty word over A to each arrow and each loop of a connected directed graph (digraph, for short). Each word is written according to the direction of its corresponding arrow or loop. Graphical sentences can be used to encode sets of sentences in a compact way: the readable sentences of a graphical sentence being the sentences corresponding to directed paths in the digraph. We apply combinatorial equations on enriched trees and rooted trees, in the context of combinatorial species and Pólya theories, to analyze parameters in classes of tree-like sentences. These are graphical sentences constructed on tree-like digraphs.


Introduction
Figure 1 (left) shows a completely unlabelled1 connected digraph.We define a graphical sentence over a finite alphabet A by attaching a non-empty word over A to each arrow and each loop of a completely unlabelled connected digraph.Each word must be written according to the direction of its corresponding arrow or loop, from source to target.Figure 1 (middle) shows a graphical sentence over alphabet {A, C, G, T} and Figure 1 (right) shows another over alphabet .Graphical sentences can be used to encode sets of ordinary sentences in a compact way: The readable sentences of a graphical sentence being the sentences corresponding to directed paths in its digraph.For example, TTT C GCCTG CAT CAT GCAATT, is a readable sentence arising from the graphical sentence of Figure 1 (middle).
In the present paper we focus our attention on the structure of graphical sentences as combinatorial objects using methods from the theory of combinatorial species [1] [2] and classical Pólya theory [3].We leave aside the generation of the readable sentences of a graphical sentence since this is easily done via the computation of powers of incidence matrices2 .Of course, special sentences among the readable sentences can be selected by adding extra structure to graphical sentences (such as source points, sink points, STOP points, counters, extensions of the alphabet by adding special characters such as , !, ?, etc).We also leave aside this aspect in our analysis of graphical sentences.
Various descriptive parameters can be attached to each graphical sentence over a given alphabet A. For example, the graphical sentence of Figure 1 (middle) is made of 7 vertices, 11 arrows, 2 loops, 52 letters, letter A appears 13 times, letter C appears 9 times, letter G appears 10 times and letter T appears 20 times.
As usual in enumerative combinatorics, families of parameters associated to structures are conveniently encoded by weight-monomials.
where ( ) a s ν is the number of occurrences of letter a in s.
In (1), each letter a A ∈ is reinterpreted as a formal variable.For example, the weight of the graphical sentence s of Figure 1 (middle) is given by ( ) 7 11 2 52 13 9 10 20 .

s x y z t A C G T = w
(2) Definition 1.2.Let  be any class of totally unlabelled connected digraphs and { } 1, 2,3, . Let S be the (countable) set of all graphical sentences over alphabet A arising from digraphs in  , the word on each arrow or loop having a length ∈  .The inventory of S is the formal sum of the weights of all graphical sentences in S : , , , , .
a A s x y z t a s As usual in enumeration problems, the (explicit or recursive) computation of an inventory of a class of structures provides a great deal of information about the structures to which it is associated.This information is extracted from the inventory through expansion, collection of terms, specialization/confluence of variables, algebraic/differential manipulations and coefficient extraction.
For example, in the present situation, expanding and collecting terms in (3) gives, of course, ( , , , ,  , , , ,   , , , , , a a a A a a A m n p q m n p q a A m n p q x y z t a c x y z t a ν ν ν where the coefficient ( ) , , , , a a A m n p q c ν ∈ is the total number of graphical sentences s ∈S having m vertices, n arrows, p loops, a total number of q letters, a ν of which are letter a, for each a A ∈ .Assigning the value 1 to each letter a and collecting terms gives ( ) (  ) , , ,  , , ,   , , , , 1 , m n p q m n p q a A m n p q x y z t b x y z t where , , , m n p q b is the number of s ∈ S having m vertices, n arrows, p loops and q letters.Letting 1 x y z = = = , gives ( ) (  ) where ( ) is the number of s ∈ S made of q letters occurring with frequencies ( ) and assigning the value 1 to each letter a gives ( ) ( ) 1,1,1, , 1 , q q a A q t e t ∈ = ∑ wS (7)   where q e is the number of s ∈ S made of q letters.Let , p q f be the number of graphical sentences s ∈ S made of p words (i.e., p is the total number of arrows and loops in s) and q letters.Then ( ) (  ) , , , 1 .
p q p q a A p q z z t f z t Moreover, if we let 1 t = in (8) and if S is a finite set, then is a polynomial, where p h is the number of s ∈ S made of p words.Differentiation gives Expected number of words in a random .1, , ,1, 1 (10)   Of course, a variety of other similar manipulations of the inventory ( ) ( ) , , , , a A x y z t a ∈ S are possible.In Section 2 we apply methods from the theory of species and Pólya theory, to express inventories of general classes of graphical sentences in terms of cycle index series.Section 3 deals with specific classes of graphical sentences: linear sentences (corresponding to path-like digraphs) and general tree-like sentences (corresponding to classes of tree-like digraphs).We conclude (Section 4) by giving suggestions for possible extensions and generalizations of our results.Various explicit examples are given and to make the text easier to read, the proofs of the main results are collected in Section 5.In a previous paper, [4], we studied the distribution of runs in arborescent words.We assume that the reader is familiar with Pólya theory [3] and with the basic concepts of the theory of combinatorial species [1]  [2].

Inventory of Graphical Sentences via Cycle Index Series
In order to give a rigorous meaning to the notion of a totally unlabelled digraph and to be able to take into account the possible symmetries within graphical sentences, we must recall first some definitions concerning labelled digraphs.A digraph on (or labelled by) a finite set V of vertices, a finite set 1 V of arrows, and a finite set 0 V of loops is an ordered pair , g g g = of injections, ( ) where ∈ and any ordered pair of distinct vertices ( ) ( ) , g j v w = means that arrow 1 j is going from v to w in the digraph g.In other words ( ) ( ) ( ) Figure 2(a) shows a digraph on the sets of loops, with loop 0 1 at vertex 4 and loop 0 2 at vertex 1.Let , g g g = be a digraph on ( ) If g g′ = then θ is called an automorphism (or symmetry) of the digraph g.For example, the triple of permutations ( ) A A A A + = + + + , be the set of non-empty words over A.
A graphical sentence over A is an equivalence class, s, of ordered triples, ( ) , , g σ σ , where g is a connected labelled digraph on ( ) are arbitrary functions assigning a non-empty word to each loop and each arrow of g.Two such triples ( ) , , g σ σ and ( ) , , g σ σ ′ ′ ′ being equivalent if there exists an isomorphism ( ) . We write ( ) to mean that s is a graphical sentence with representative ( ) , , g σ σ .Now take any species  of connected labelled digraphs 4 .Our goal is to compute the inventory (3) of the class S of all graphical sentences ( )  where g ∈ .To emphasize the fact that digraphs are made of three sorts of elements, vertices, arrows and loops, the given species  of digraphs can be written in the form (  ) , where X is the sort of vertices, Y is the sort of arrows, and Z is the sort of loops 5 .Any digraph g ∈ is called a  -structure for short.
Following standard notations from the theory of species, the set of all  -structures on a set V of vertices, a set 1 V of arrows and a set 0 , , V V V  (note the square brackets).Given bijections the bijection ( ) Many power series can be associated to any species  .An important one is the Pólya-Joyal cycle index series Z  .In the context of a species ( ) of digraphs, this is a power series in a triple infinity of variables, 1 2 3 , , , ; , , , ; , , , x x x y y y z z z   , defined by ( ) , , , ; , , , ; , , , [ ] ).For a permutation k S σ ∈ , the notation i σ is used to de- note the number of cycles 7 of length i in the cyclic decomposition of σ .
The sequence of integers ( ) , , , σ σ σ  is called the cyclic type of σ and it is well known that the number of k S σ ∈ having cyclic type ( ) Note that each sequence k has a finite number of nonzero terms and will be considered, in the present text, as a finite sequence with  ( ) ( ) x y z t a 6 Or total weight, in the case of weighted digraphs. 7Not to be confused with ( ) is the formal k-th power sum of the letters of the alphabet.

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Making use of the compact expression (22) for the cycle index series and collecting terms, inventory (23) can be rewritten in the following more explicit form.
and no restrictions are put on the lengths of the words in inventory (23 and the lengths of the words are all odd.If and the lengths of the words are bounded by N, etc.

Analysis of Classes of Tree-Like Sentences
As shown in the preceding section, the computation of the inventory of a class S of graphical sentences can be reduced to the computation of the cycle index series Z  provided that S arises from a 3-sort species ( ) of connected digraphs.However, the explicit or recursive computation of the cycle index series of most species of graphical structures is a very difficult (or intractable) task.For example, even in the ordinary one-sort case 9 , the complete cycle index series of the species of all ordinary plane digraphs and all transitive digraphs are still unknown 10 .
For this reason, we focus our study on the following basic classes of graphical sentences: 1) Linear sentences (arising from the species of path-shaped digraphs).
2) General tree-like sentences (arising from various species of tree-like digraphs).Note that linear sentences are special kinds of tree-like sentences.Due to their close relationship with ordinary sentences, we have chosen to present first a separate subsection devoted to their study.Our methods will use the fact that species of tree-like digraphs can be built from simpler species by making use of basic combinatorial operations and that cycle index series behave well with respect to these operations.For example, if F, G and H are species, then where  denotes the classical plethystic substitution of cycle index series (see [1]).

Linear Sentences
We say that a digraph ( ) is path-shaped if its underlying simple graph is a simple path.A graphical sentence is linear if it comes from a path-shaped digraph.Figure 3 shows a path-shaped digraph, together with its underlying simple path and a linear sentence over alphabet . Note that a path-shaped digraph can have non-trivial automorphisms.For example, the 180˚ rotation, (  )   , , α β γ , where the cyclic decompositions of the permutations , , α β γ are given by is an automorphism of the path-shaped digraph of Figure 3. 9 Where only the vertices are labelled. 10However, for the 3-sort species

( )
Dig , , X Y Z of all digraphs, Z Dig can be computed explicitly (see Section 4).
Special kinds of linear sentences over an alphabet include ordinary sentences (Figure 4 top), corresponding to directed paths without loops, and ordinary sentences with (possible) loops (Figure 4 bottom), corresponding to directed paths with (possible) loops.
For example, the sentence MY TAYLOR IS RICH RICH AND MY COUSIN IS POOR POOR POOR (29) is one of the readable sentences in Figure 4 bottom.Proposition 3.1 Let  be the set of all ordinary sentences,   , the set of all ordinary sentences with loops, L , the set of all linear sentences without loops, and L  , the set of all linear sentences with loops over an al- phabet A and a set + ⊆   of allowed word-lengths.Then, the following inventories hold ( ) ( ) where, ( ) ( ) Proof.See Section 5.

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In view of (30)-(32), the alternate general inventory formula (24) in the case of any class of linear sentences, does not involve 3 4  , , λ λ  .We have the following explicit expansions.

Corollary 3.2. For the classes , , , L L
    of linear sentences, we have   where using the convention 0 if α and β are not both integers.

Sample of explicit examples of computations. Example 3.1. General shape of the inventory of the class L 
w .The first few terms of L  w read as follows x y x y z x y z x y z The coefficients of x y z is given by where The required number of linear sentences equals the coefficient of 25  As we have seen in Section 1, manipulations of inventories (specialization of variables, expansions, etc) can be made to analyze various parameters in graphical sentences.Proposition 3.1 is generally more suitable for such manipulations than Corollary 3.2.For example, let  be the number of letters in alphabet A, + =   , and assign the value 1 to x, y and to each letter a A ∈ in wL given by (31).Then ( ) ( ) where q e is the number of linear sentences without loops that are made of q letters.Note that (46) is a rational function of t, so that the sequence ( ) 0 q q e ≥ satisfies a linear recurrence with constant coefficients and the asymptotic expansion of q e , as q → ∞ , can be established using standard classical methods.The first few terms in expansion (46) are given by  given by (30).Then we have ( ) ( ) where k F are the Fibonacci numbers 11 and is the number of ordinary sentences with possible loops made of p words and q letters.Now, fix 1 q ≥ and consider the finite class q   of all s ∈   made of q letters.Since, ( )  , then collecting the coefficient of q t in (48) we have the polynomial inventory Finally, making use of (10) and invoking Binet's formula ( ) is the golden number, , we find that the expected number of words in a random ordinary sentence with possible loops made of 1 q ≥ letters is The reader can check that if we do not allow loops, then (51) is replaced by ( ) A multitude of other similar examples can be obtained using Proposition 3.1 and Corollary 3.2.

General Tree-Like Sentences
We say that a digraph ( ) is tree-like if its underlying simple graph is a simple tree or a simple rooted tree.A graphical sentence is tree-like if it comes from a tree-like digraph.Figure 5 shows a tree-like digraph, its underlying simple tree and a tree-like sentence over alphabet The tree-like structures of Figure 5 are free in the sense that they are not restricted to be embedded in the plane and no other constraints are assumed on the vertices, arrows and loops.More generally, by allowing such constraints, one can consider, for example, the above linear sentences (see Figure 3 and Figure 4), one way free binary rooted tree sentences (see Figure 6 left), one way free full binary rooted tree sentences (see Figure 6 right), plane tree sentences (see Figure 5 right) where, this time, the underlying tree is considered as being embedded in the plane), etc.We shall deal with these cases in a uniform manner by adding extra structure on the underlying trees or rooted trees.More precisely, the underlying trees or rooted trees will be enriched according to the following definition.
Definition 3.1.[5] Let be any given one-sort species.1) A R-enriched rooted tree is a rooted tree in which the set of immediate descendants (away from the root) of every vertex is equipped with a R-structure (see Figure 7 left, in which each dotted arc represents a R-structure).2) A R-enriched tree is a tree in which the set of immediate neighbors of each vertex is equipped with a R-structure (see Figure 7 right, in which each dotted circle represents a R-structure).
Lemma 3.3 [1] The species and the species where is the species of R'-enriched rooted trees (R' being the combinatorial derivative of the species R).

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It is easy to see that the species of ordinary rooted trees (resp.ordinary trees) corresponds to the species A R (resp.a R ) with the choice R E = , the species of all finite sets.The species of binary rooted trees (resp.full bi- nary rooted trees) corresponds to the species A R with the choice , where 1 denotes, as usual, the species of the empty set and E 2 , the species of 2-element sets.The species of all plane trees corresponds to the species a R with the choice 1 R C = + , where C is the species of cyclic permutations (see Example 3.8 below), etc.
For the computation of the inventories of various classes of enriched tree-like graphical sentences, we will make use of the following 3-sort extension of Lemma 3.3 which includes a new extension, (56) below, of the dissymmetry formula (53).

R-enriched rooted trees on the sorts X of vertices, Y of arrows and Z of loops are characterized recursively by the combinatorial equations
where They can also be expressed explicitly in terms of the 1-sort species The species ( ) of R-enriched trees on sorts X of vertices, Y of arrows and Z of loops satisfies the combinatorial equality (extended dissymmetry formula) is the species of R'-enriched rooted trees on sorts X, Y, Z. Proof.See Section 5.

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In our analysis of tree-like graphical sentences, we will use of the following useful compact "plethystic nota-tion" which is classical in the theory of species and cycle index series.Notation 3.5.Let ( )  be a (formal) power series in the variables 1 2 3 , , , v v v .For any integer 1 k ≥ , k S denotes the series S in which each variable is raised to the power k: ( ) ( ) , , , , , , .
In particular, 1 = S S. Furthermore, given power series, ( ) ( ) , , , , , , , ; , , , , , s s s s s s They can also be expressed explicitly in terms of the cycle index series ( ) where ( ) Moreover, let R' be the combinatorial derivative of the species R.Then, the inventory In the case of the corresponding sets, R  T , R T , R t , in which no loops are allowed, we have where , ⋅⋅⋅, ( ) 65) and (66) give rise to iterative schemes for the computation of the inven- tories wT  .See, for example, [7]- [10] for descriptions of efficient ways to do such computations, in- cluding adaptations of quadratically convergent Newtonian methods.Formula (69) reduces the computation of R wt  to that of R′ wT  .Recall that ( ) ( )

Sample of explicit examples of computations.
Example 3.5.One way free binary rooted tree sentences.
Consider the set  B of one way binary rooted tree sentences without loops (Figure 6 left, shows such a tree-like sentence over the 26-letter alphabet

{ }
a,b, ,z A =  ) and the set  B  of such sentences where loops are allowed.These tree-like sentences correspond to R-enriched rooted trees with 65) of Proposition 3.6 immediately gives the following recursive scheme for the computation of the inventory w and, since Of course, as many terms as we want in ( 72) and ( 73) can be computed using a computer algebra system.For a more specific application, let  be the number of letters in alphabet A and let and ( ) ( ) ( ) as in (7).Then ( ) The coefficient of q t in (75) (resp.( 77)) is the number of one way free binary rooted tree sentences with loops (resp.without loops) on a  -letter alphabet A that are made of q letters.As an illustration, for the usual 26-letter alphabet

{ }
a,b, ,z A =  , series (75) and (77) read as follows up to 10 q = : ( ) , be the species of finite sets.Then, by Definition 3.1, the species ( ) , the second equation being the classical dissymmetry formula of Leroux.Taking cycle index series in (80) and using R E ′ ′ = = and ( ) , we obtain the classical formulas ( ) ( ) ( ) , , , , , , a Z x x x  can be computed to arbitrary degree 13 .

Now, let
be the set of one way free rooted tree sentences with possible loops,  T  (85) be the set of free rooted tree sentences with possible loops, T  (86) be the set of free tree sentences with possible loops, t  (87) Then, by (67)-(69), ( ) For more specific applications, let  be the number of letters in the alphabet and consider the specializations  , and by (66) and (69) with R E = , we have The coefficient of q t in (92) (resp.( 94)) is the number of free rooted tree sentences (resp.free tree sentences) with loops on a  -letter alphabet A that are made of q letters.For example, in the case of a 4-letter alphabet, say , series (92) and (94) read as follows up to 9 q = : ( ) , and make the substitutions where , i j g (resp., i j h ) is the number of rooted tree sentences (respected tree sentences) without loops that contain exactly i times the letter a and j times the letter b.
If we choose { }  Since linear sentences are special kinds of tree-like sentences, it is interesting to look at the dissymmetry formula (56) in the context of path-shaped graphs.Take the 1-sort species . Then a R-structure is either void, a singleton, or an unordered pair of singletons.This means that a R-enriched tree is a simple path (see Definition 3.1, Figure 7 right and Figure 3 middle).Hence, the 2-sort species ( ) , P X Y of all path-shaped digraphs without loops (see Figure 9) coincides with the 2-sort species  of R'-enriched rooted trees.In this setting, the dissymmetry formula (56), with 0 Z = , becomes where ( ) ⋅ + Ω , we can solve (101) for P as follows, This formula coincides with formula (124) which is used in the proof of Proposition 3.1.

Example 3.8. Plane tree sentences.
A plane tree is a (unrooted) tree that is embedded in a plane.Such tree-structures have fewer automorphisms than free trees.Take any vertex p of a plane tree τ and draw a vector starting at p which is perpendicular to the plane in which τ is embedded.This gives an orientation to that plane and the vertices that are adjacent to p are cyclically turning around p according to that orientation (see Figure 7).In other words, the set of immediate neighbors of p is equipped with a ( ) where C is the species of non-empty oriented cycles (the empty set species, 1, corresponds to the special case where the tree is reduced to one point, , for which the the set of immediate neighbors of p is empty).Since p is arbitrary, this shows that the species plane a of plane trees coincides with the species 1 C a + of ( ) = is the species of linear orders.The species L A of Lenriched rooted trees coincides with the species o A of linearly ordered rooted trees (the set of immediate descendants, away from the root, of every vertex is linearly ordered).Using the classical formulas, ( ) ) Using the expansion 14 ( ) This time, the coefficient of q t in (109) (resp.( 110)) is the number of linearly ordered rooted tree sentences (resp.plane tree sentences) with loops on a  -letter alphabet A that are made of q letters.As a final illustration, fix 1 m ≥ and consider the inventory Now, let  be the number of letters in the alphabet and assume that the length of the word on each arrow is at most k.Then, making the substitutions, which is a polynomial in t, since ( ) where , , m k e  is the expected total number of letters in random m-vertex linearly ordered rooted tree sentence without loops on a  -letter alphabet in which the word on each arrow has at most k letters.Further computa- tions give , , 1 Again, all the above inventories can be manipulated in a great number of ways.

Concluding Remarks
It would be interesting to extend the above analysis to other classes of graphical sentences arising from other and, since , , , Proposition 2.1 immediately gives (30).On the other hand, the inventories of the sets L and L  of linear graphical sentences are more difficult to compute since a path-shaped digraph can have a nontrivial automorphism (as we saw above).We first analyze the species ( ) , P X Y of all path-shaped digraphs without loops (see Figure 9 top).Introduce the auxiliary spe- cies be the species of all P-structures pointed at an extremity (see Figure 9 bottom).This pointing induces a global orientation to these pointed structures (see dotted arrow) and implies that the species K is a species of sequences: As a consequence of the general dissymmetry formula (56) the species P can be expressed in terms of K and Ω as follows (see details in Example 3.7) where 2 E denotes the species of 2-element sets.Formula (31) then follows from Proposition 2.1 by taking the cycle index series of (124) and using the fact that . Finally, let ( ) be the species of all path-shaped digraphs possible with loops, then the following combinatorial equation holds since every P  -structure is obtained from a P-structure by adding a loop to each vertex (that is, : X ZX = ) or doing nothing to the vertex (that is, : X X = ).So that (32) follows by substituting ( )  arrow (that is, by replacing each such edge by an Y-structure).This establishes (54a).The proof of the combinatorial Equation (54b) is similar, where, this time, each edge adjacent to the root of t is replaced by an outward arrow, an inward arrow or a double arrow (that is, by replacing each such edge by a ( ) 2Y Y Ω = + -structure).To obtain the explicit formula (55a), multiply first both sides of (54a) by Y.This gives ( ) ( ) But, by (52), the species ( ) ( ) also satisfies (126).Hence, by the unicity of solution in the species Theorem of Joyal [2], we must have ( ) (  ) , and (55a) follows by factoring out Y. A similar argumentation can be used to prove (55b) from (54b).
The dissymmetry formula ( 56) is much more difficult to establish since more automorphisms are involved in enriched trees.To prove this combinatorial equality, we express in two ways the auxiliary species ( ) a  -structures which are pointed either at a single vertex or at two adjacent vertices: • The first expression for R  reads as follows ( ) ( ) ( ) ( ) ( ) To prove it, consider a R  -structure φ and look at its underlying pointed or bipointed R-enriched tree, f.
We have two cases to consider:  -structure since to recover φ from f, the vertices of f must be replaced by ( ) -structures and the edge adjacent to the pointed vertex (and subsequently, all other edges) must be replaced by an Ω-structure.
To prove it, we first split the species R  into two subspecies according to whether the pointing(s) coincides exactly with the center or not: Since the center of a tree is a canonical object, pointing a tree exactly at its center is naturally equivalent to doing nothing to the tree and we have    ≠ .Indeed, the bi-pointing induces an orientation on the edge between the pointed vertices of f in the direction opposite to the center (see dotted arrow) giving rise to an ordered pair of rooted trees.To recover φ from f, that edge must be replaced, as above, by an Ω-structure.Note that 2 φ is not an arbitrary ( ) ( )

Definition 1 . 1 .
The weight of a graphical sentence s over an alphabet A is the (commutative) formal monomial3

Figure 3 .
Figure 3. Path-shaped digraph, its underlying simple path and a linear sentence.

Figure 4 .
Figure 4.An ordinary sentence and an ordinary sentence with loops.

2 pExample 3 . 2 .
in (41) are polynomials in 1 λ and 2 λ each having one or two terms, despite the fact that (35) suggests three terms.This is true for every m, n, p since , , cannot be both integral in (39) and (40).Counting linear sentences with given parameters.Corollary 3.2 is particularly useful when one wants to compute an individual term in the inventory of linear sentences.For example, consider the 3-letter alphabet { } a,b,c A = and take + =   .This means that we impose no restrictions on the lengths of the words that are assigned to each arrow or loop in linear sentences.Suppose that we want to know the number of such linear sentences having 7 m = vertices, 10 n = arrows, 5 p = loops which are made of 7 times the letter a, 6 times the letter b and 12 times the letter c.In this case, the coefficient of 7 10 5

47) Example 3 . 4 .
Fibonacci numbers versus ordinary sentences with loops.Consider now the class   of ordinary sentences with possible loops.Let y z = and assign the value 1 to x and to each letter a A ∈ in   w

Figure 6 .
Figure 6.A one way free binary rooted tree sentence and a full one.

Figure 7 .
Figure 7.A R-enriched rooted tree and a R-enriched tree.
is the formal sum of the letters in A. We now describe how to compute the inventory of classes of tree-like sentences.Proposition 3.6 Given an arbitrary species( ) R R X = , letbe the set of one way -enriched rooted trees sentences with possible loops, of -enriched rooted trees sentences with possible loops, of -enriched trees sentences with possible loops, allowed word-lengths.Then, using Notation 3.5, the inventories R 70) Proof.(sketch) Apply Proposition 2.1, taking into account Lemma 3.4.Note.When written explicitly, (65) takes the form

(
enriched rooted trees coincides with the species ( ) A A X = of ordinary (free) rooted trees and the species ( ) E a X of E-enriched trees coincides with the species ( ) a a X = of ordinary (free) trees.Lemma 3.3 produces the familiar combinatorial equations, ( are allowed, and (89)-(90) take the forms rooted tree is a simple path pointed at an extremity (see Definition 3.1 and Figure7, left).Hence, the 2-sort species ( ) , K X Y of all path-shaped digraphs without loops pointed at an extremity (see Figure9) coincides with the 2-sort species of linearly ordered rooted tree sentences without loops having exactly m vertices.Letting 0 z = in (105) and (108), we have

1 )(
If φ is a ( ) R • a  -structure, then by Figure 10 we see that φ is canonically equivalent to a

2 )(
If φ is a ( ) then Figure11shows that φ is canonically equivalent to a to recover φ from f, the edge of f between the two adjacent pointed vertices must be replaced either by a double arrow (i.e., φ is equivalent to a -structure) or by a sin- gle arrow (i.e., φ is equivalent to a ( ) .This establishes (128).•The second expression for R  reads as follows ( ) ( )

 1 φ
-structure φ that is not pointed at its center.We have two cases to consider:1) If φ is an ( ) center R •≠ a -structure, then Figure12shows that φ is canonically equivalent to a .Indeed, the pointing induces an orientation on the edge from the pointed vertex of f in the direction of the center (see dotted arrow) giving rise to an ordered pair of rooted trees.Moreover, to

Figure
Figure Underlying structures for ( ) { } since the global center is now on the side of the source of the dotted arrow.This establishes (129) since any ( either of the form 1 φ or of the form 2 φ .The general 3-sort dissymmetry formula (56) follows by cancelleing the common term in the right-handsides of the expressions (128) and (129) for R  .

Ordinary tree and rooted tree sentences. Let
.