Combinatorial Interpretation of Raney Numbers and Tree Enumerations

A new combinatorial interpretation of Raney numbers is proposed. We apply this combinatorial interpretation to solve several tree enumeration counting problems. Further a generalized Catalan triangle is introduced and some of its properties are proved.


Introduction
Interestingly Penson and Zyczkowski were the first who use the term Raney numbers [1] [2] and it is defined as ( ) Nevertheless, it is known that Raney's lemma could be used in counting problem associated with Catalan numbers [3] and a bijection exists between Raney path and plan multitree [4].These numbers do not form novel sequences, as the numbers were introduced earlier as a generalization of the binomial series [5].Moreover, the sequence ( ) 4 , 5 1,5,30, 200,1425,10626,81900, 647280, R n =  is not included in OEIS database [6] before 2011.If we let 1 r = , we obtain another known sequence, i.e., Fuss-Catalan numbers [7] [8] which is defined as ( ) ( ) . Although Fuss-Catalan numbers were introduced earlier than Catalan numbers [9], the Catalan numbers are more popular and widely used than the Fuss-Catalan numbers (see [10] [11] for details).Due to its self similar structure, the applications of Catalan numbers could be found in many physical problems, e.g., lattice model [12], tree enumeration network [13], and Hankel matrices in coding theory [14].A tree is a connected graph with no cycles and for which only one shortest path exists from one node to another.Tree enumeration is an important tool to study network.These networks always grow in a power-law behavior which is often found in social network, subway system [15], etc.
In this paper, we introduce Raney numbers ( ) , R k r in the form of a non-linear recursion and then we pro- vide a combinatorial interpretation of Raney numbers.Using this combinatorial interpretation, we solve several tree enumeration counting problems in which we recover the well-known Fuss-Catalan numbers [16], Catalan triangles [17], and other less known numbers.Motivated by the connection between Raney numbers and Catalan triangles, a generalization of Catalan triangles is proposed and we prove some of their properties.Consequently these formulas generalize the properties of Catalan triangles.From the exact solution of these tree enumeration problems, we are able to find a sharp upper bound of the number of each tree enumeration problem.The upper bound is important in the contour method for lattice models and limit of the random graph.

Raney Numbers
Let ( ) k C n be the number of a k -ary trees with labeled n vertices (Figure 1), where ( ) ( ) The Raney numbers are defined as follows: where , , , 0 . Therefore, the combinatorial interpretation is as follows: r copies of k -ary tree with total number of n vertices.
Next, we let ( ) k u x be the generating function for . n r kn r r x u x n kn r Immediately, we obtain the following theorem.Theorem 1.The binomial forms of the Raney numbers are given by ( ) From theorem (1), it is not difficult to deduce some of the properties of Raney numbers.Corollary 1.For integer 1 k > , we have ( ) Figure 1.A binary tree with 3 nodes, where the bottom vertex is the root.

( )
, ; ( ) Corollary 2. We can write C n in a nonlinear recursion as: where , , , 0 . We recover the formula by joining the k copies of k -ary tree with n vertices which is also equivalent to a k -ary tree with n vertices and an additional root (see Figure 2): Using binomial form of ( ) R n r , one can obtain the following result.Corollary 3.For a fixed integer 1 k > , and where ( ) we recover the identity of a generalized Ballot numbers:

A Homogeneous k-Ary Tree
Unlike the usual k -ary tree, we define a homogeneous k -ary tree as a graph with no cycles, in which each vertex emanates 1 k + edges (see Figure 3 for k = 4).We fix a vertex namely z as the root.Unlike the ordinary root in a k -ary tree, this root has 1 k + successors while other vertices have k number of successors.Any vertex could be chosen to be the root since the graph is homogenous.For a given n vertices, we may find how many connected sub-tree rooted at z .This number is defined as Proof.We decompose the problem by finding out the number of k -ary tree of n number of one copy of k -ary tree with r vertices, i.e.,   Since the former r C must always include 0 x , its range should be from 1 to n .Total ( ) C r C n r − using the addition and multiplication principles.Using Equation ( 8), we rewrite the formula above as This formula can also be obtained using 1 k + copies of k -ary tree together with 1 n − vertices and one center.
From theorem (2), one can get This formula can be also obtained easily by a different way: 1) Count the number of trees by joining the 2 copies of ary k n − tree, with total number of vertices n .2) Subtract those trees remunerate from y but doesn't contain z , that is, exactly the number where ( ) .
Using the binomial inequality in [21] and the binomial forms of ( ) ( ) k D n , the following in- equality can be easily proved.
Corollary 7.For where For sufficiently large k , a simpler form is produced as well, i.e., ( ) ( ) ( ) . These results are conjectured in a weaker form in [20], i.e.,

Catalan Triangle
A Catalan triangle ( ) , B n r is defined as follows [9]: The Catalan triangle satisfies [9]: Using a property of Catalan numbers, ( ) ( ) ( ) , where 1 2 , 0 j j ≥ , we get another form of ( ) , , r r j j j n r B n r C j C j C j where , , , 0 . From Equation (1), we immediately recover the Catalan triangle from the Raney numbers, i.e., ( ) ( ) We now consider the following problem as in [22]: Find out the number of all different connected sub-trees of a homogenous binary tree with n number of vertices, containing the given r number of fixed vertices (where 2 2 n r ≥ − ).The condition, 2 2 n r ≥ − , is simply the number of vertices that covers the minimal component containing all r vertices.The details of this problem and terminologies could be found in the original paper [22].We denote the solution to this problem as r n F .In this paper, we show that a solution to the case when the minimal component is "full", is as below: where n is the number of given vertices and r is the number of fixed vertices in each of the connected sub- tree.Now, we interpret and relate the problem above with the combinatorial interpretation of the Raney numbers through the following steps (see Figure 4): 1) Given n vertices; 2) Fill up all the interior points, i.e., 2 r − ; 3) Fill up all the boundary points, i.e., r ; 4) Then only 2 2 n r − + vertices are left; 5) Since each boundary point has 2 neighbours which is not an interior point, we have 2r boxes; 6) If 2 2 n r − + vertices are given, then there are 2r boxes of binary tree to be filled.As a result, the solution is Furthermore, it is natural to define a generalized Catalan triangle, i.e., k -th Catalan triangle using Fuss-Catalan numbers instead of Catalan numbers as in Equation ( 19): where ( ) From the property of Fuss-Catalan numbers, i.e., corollary ( , we find another form of ( ) , , where , , , 0 . Again, from Equation (1), we immediately have Lemma 2. Some properties of k -th Catalan triangles are as follows: Using the binomial form of ( ) Based on the initial result, lemma (3), we prove the following assertion by mathematical induction with respect to m .Theorem 3.For fixed k , where 1 k m ≥ > , and where ( ) Assertion is true for 2 m = .Assume that it is true for m , we consider the following summations: Hence, the assertion is true for any For m k = , we have the following simple result: Corollary 9.For fixed 1 k > , and

Binomial Transformation of k-th Catalan Triangle
 , all the Fuss-Catalan numbers should start at 1, then we recover the previously defined k -th Catalan triangle.

Conclusion
In this paper, we have introduced the combinatorial interpretation of Raney numbers to solve various tree enumeration counting problems.The upper bound of any 1 k + order tree enumeration is generally found to be ( ) the generating function of the Raney numbers.Then, ( and another copy of k -ary tree with n r − vertices, i.e.

Figure 3 .
Figure 3.A homogenous graph where each vertex is connected to exactly 5 neighbours.
Then we have the following result.Corollary 5.For 1 k > and 0 n ≥ , the generating function, ( )
recover the same formula for k -th Catalan triangle, as a result of binomial transformation of k -Catalan triangles.= if n r < .
derived from the binomial transformation of k -th Catalan triangles.