BURGARD@DBNINF5.BITNET (11/21/90)
I am interested in the following problem: What is the order of the average depth of the leafs of full binary trees with n leafs? Remarks: - A full binary tree is a binary tree with each node having outdegree either 2 or 0 - The number of full binary trees with n leafs is the Catalan number C(n-1) - C(n)= 2n \over n / (n+1) - The average depth A(n,k) of the leafs of a full binary tree with n leafs and k leafs in the left subtree of the root is: A(n,k)= 1 + ( k*A(k) + (n-k)*A(n-k) ) / n - Thus the average depth A(n) of the leafs of a full binary tree with n leafs is: A(n)= 1+ \sum_{k=1}^{n-1} { f(n,k) * k*A(k) + (n-k)*A(n-k) ) } / n where f(n,k)= C(k-1)*C(n-1-k)/C(n-1). - trivial bounds for A(n): O(log(n)) \leq A(n) \leq O(n) - Conjecture: A(n) is sublinear, e.g. O(n^c) with c<1 Are there any results concerning this problem? Thanks for the interest, -- Wolfram Wolfram Burgard Institut fuer Informatik III Universitaet Bonn BURGARD@DBNINF5.BITNET
nachum@m.cs.uiuc.edu (11/22/90)
/* Written 4:21 pm Nov 20, 1990 by BURGARD@DBNINF5.BITNET in m.cs.uiuc.edu:comp.theory */ /* ---------- "Full Binary Trees" ---------- */ I am interested in the following problem: What is the order of the average depth of the leafs of full binary trees with n leafs? ... - Conjecture: A(n) is sublinear, e.g. O(n^c) with c<1 ... /* End of text from m.cs.uiuc.edu:comp.theory */ Your conjecture is correct. It is O(sqrt(n)), by results in Knuth's vol. 1, sect. 2.3.4.5 on "external path length". (See also my paper with S. Zaks, "Patterns in Trees", in Discrete Applied Math, vol. 25, pp. 241ff., result 3.5.1.)