The Faulty Tree

Alice and Bob will play a game on an array $A$, consisting of $N$ numbers. The rules of the games are: - Alice and Bob will construct a binary tree of exactly $N$ leaves. - Each node of the tree is given a value: - If the node is a leaf, its value will be one of the unused array's elements. - Else, the node's value will be the sum of its two children. - The ugliness of the tree is defined