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A simple proof of a formula of Dershowitz and Zaks
Discrete Mathematics43 (1983) 117-118 North-HollandPddishing Company
NOTE A SWPLE PROOF OF A FORMULA OF D~RSHOWITZ AND ZAKS Frank RUSKEF Department of Computer Science, University of Victoria, Vktoria, B.C. VSW 2Y2 Canada Received 16 September 1981 R&sed 19 February 1982 Let J&(1,d) be the number of nodes in the set of all ordered trees with n edges that are at level I and have degree d. Dershowitz and Zaks [l] prove that
We reprove this result by showing an explicit correspondence between nodes in trees with n edges that are at level I and have degree d and trees with n + I edges and rlstit degree 21+ d.
Let 5,, be the set of ordered trees with it edges and let V,, be the number of ordered trees with n nodes. The numbers %*, Q2, . . . are the Catalan numbers. Following Dershowitz and Zaks [ 11, define 411,(r) to be the number of trees in 9,, with root degree r, and &,( 1,d) to be the total number of nodes of degree d at level I summed over all trees in $,,. It is known that
The first equality is obvious; several proofs of the second have appeared [1,3,4]. In [l] it is shown that
JW, 4 =-
(2)
Their proof finds a defining set of recurrence relations and boundary conditions that the &,(I, d) satisfy, and then verifies that the numbers in the right hand side of (2) satisfy them also. A second proof, by lattice path techniques, was given by these same authors in [2]. Here we observe that &,(I, d) = $l$,+,(21+ d) and try to establish tlhe correspon.dence suggested by the equality. The proof given here, together with the proof of * Research supported by NSERC grant A-3379.
0012-365X/83/0000-0000/$03.00
@ 1983 North-Holland
F. l&key
118
(1) in [ 11, results in a very short proof of (2). Furthermore, the proof is entirely by correspondences, and does not rely on induction or generating functions. Consider a generic node of degree d at level I of a tree in S,, as shown ii: Fig. l(a). The shaded triangles represent forests and the unshaded triangles arc SXXS. There are %n+l ordered forests with tl nodes.
I-
(!:I
Fig. 1. The correspondence.
The correspondence is the one suggested by Fig. l(b). The 21 forests are rooted by introducing I additional nodes and using the 1 nodes on the path to the generic node. In Fig. l(b) the new nodes are blackened. The generic node is made the root and the forests (which have been madt- into trees) are added as subtrees of the root. The whole process is cJearly reversible. One can alSo proceed as follows:
Oerc r; is the number of nodes in the ith forest and pi is the number of nodes in the 13 subtree (of the generic node). Here the idea is the same but the correspondence is not made explicit. References [ 11 N. Denhowitz and S. Zaks. Enumerations of ordered trees, Discrete Math. 31 (IYSO) 9-28. [2] N. Derr,howitz and S. Zaks Applied tree enumerations. 6th CAAP, Genoa, Lecture Notes in Computer Science 112 (Sprir;ger-Vcrlag, Berlin, 1% 1) I W- 193. [3] F. Ruskey and T.C. Hu, Generating binary trees lexictrgraphically, SIAM J. Comput. 6 (1977) 73.5-758. I41 D.W. Walkup. The number of plane trees, Acta Math. 19 (1972) 200-204.
NOTE A SWPLE PROOF OF A FORMULA OF D~RSHOWITZ AND ZAKS Frank RUSKEF Department of Computer Science, University of Victoria, Vktoria, B.C. VSW 2Y2 Canada Received 16 September 1981 R&sed 19 February 1982 Let J&(1,d) be the number of nodes in the set of all ordered trees with n edges that are at level I and have degree d. Dershowitz and Zaks [l] prove that
We reprove this result by showing an explicit correspondence between nodes in trees with n edges that are at level I and have degree d and trees with n + I edges and rlstit degree 21+ d.
Let 5,, be the set of ordered trees with it edges and let V,, be the number of ordered trees with n nodes. The numbers %*, Q2, . . . are the Catalan numbers. Following Dershowitz and Zaks [ 11, define 411,(r) to be the number of trees in 9,, with root degree r, and &,( 1,d) to be the total number of nodes of degree d at level I summed over all trees in $,,. It is known that
The first equality is obvious; several proofs of the second have appeared [1,3,4]. In [l] it is shown that
JW, 4 =-
(2)
Their proof finds a defining set of recurrence relations and boundary conditions that the &,(I, d) satisfy, and then verifies that the numbers in the right hand side of (2) satisfy them also. A second proof, by lattice path techniques, was given by these same authors in [2]. Here we observe that &,(I, d) = $l$,+,(21+ d) and try to establish tlhe correspon.dence suggested by the equality. The proof given here, together with the proof of * Research supported by NSERC grant A-3379.
0012-365X/83/0000-0000/$03.00
@ 1983 North-Holland
F. l&key
118
(1) in [ 11, results in a very short proof of (2). Furthermore, the proof is entirely by correspondences, and does not rely on induction or generating functions. Consider a generic node of degree d at level I of a tree in S,, as shown ii: Fig. l(a). The shaded triangles represent forests and the unshaded triangles arc SXXS. There are %n+l ordered forests with tl nodes.
I-
(!:I
Fig. 1. The correspondence.
The correspondence is the one suggested by Fig. l(b). The 21 forests are rooted by introducing I additional nodes and using the 1 nodes on the path to the generic node. In Fig. l(b) the new nodes are blackened. The generic node is made the root and the forests (which have been madt- into trees) are added as subtrees of the root. The whole process is cJearly reversible. One can alSo proceed as follows:
Oerc r; is the number of nodes in the ith forest and pi is the number of nodes in the 13 subtree (of the generic node). Here the idea is the same but the correspondence is not made explicit. References [ 11 N. Denhowitz and S. Zaks. Enumerations of ordered trees, Discrete Math. 31 (IYSO) 9-28. [2] N. Derr,howitz and S. Zaks Applied tree enumerations. 6th CAAP, Genoa, Lecture Notes in Computer Science 112 (Sprir;ger-Vcrlag, Berlin, 1% 1) I W- 193. [3] F. Ruskey and T.C. Hu, Generating binary trees lexictrgraphically, SIAM J. Comput. 6 (1977) 73.5-758. I41 D.W. Walkup. The number of plane trees, Acta Math. 19 (1972) 200-204.