General Computer Science 320201 GenCS I & II Lecture ... - Kwarc

Corollary 399 A fully balanced tree of depth d has 2 d+1 − 1 nodes.
Proof:
P.1 Let G := 〈V, E〉 be a fully balanced tree
Then #(V ) = d
i=1 2i = 2 d+1 − 1.
c○: Michael Kohlhase 230
This shows that balanced binary trees grow in breadth very quickly, a consequence of this is that
they are very shallow (and this compute very fast), which is the essence of the next result.
Depth Lemma for Balanced Trees
P.2 Lemma 400 Let G = 〈V, E〉 be a balanced binary tree, then dp(G) = ⌊log 2(#(V ))⌋.
Proof: by calculation
P.1 Let V ′ := V \W , where W is the set of nodes at level d = dp(G)
P.2 By the size lemma, #(V ′ ) = 2 d−1+1 − 1 = 2 d − 1
P.3 then #(V ) = 2 d − 1 + k, where k = #(W ) and (1 ≤ k ≤ 2 d )
P.4 so #(V ) = c · 2 d where c ∈ R and 1≤c