Addition palindrome sequences and trees

Gunmo was playing with sequences and palindromes when he developed addition palindrome sequences.

A sequence is an addition palindrome sequence if it satisfies the following conditions.

1. Choose two adjacent numbers in the sequence.
2. Remove the two chosen numbers, and insert their sum at the position where they were removed.
3. Repeat steps 1 and 2 as many times as you want. (You do not have to repeat them.)
4. If the final sequence is a palindrome of length 2 or more, the initial sequence is an addition palindrome sequence.

A palindrome sequence reads the same forward and backward. For example, $[11,12,13,12,11]$ and $[11,12,12,11]$ are palindrome sequences, but $[1,11,111]$ and $[1,12,21,1]$ are not.

$[1,1,3,5,3,2]$ is an addition palindrome sequence because it can be turned into $[2,3,5,3,2]$. $[1,2,5,2]$ is not, because no sequence of operations turns it into a palindrome of length 2 or more.

Gunmo thought addition palindrome sequences were too easy. So he wants to count how many simple paths between pairs of leaf nodes in a full binary tree form addition palindrome sequences. The pairs $(1,2)$ and $(2,1)$ are the same pair, so count each one once.

A full binary tree is a binary tree where every node has 0 or 2 children, and every leaf is at the same depth. In other words, every leaf is the same distance from the root.

The figure below shows a full binary tree of height $3$.

If you choose the leaf pair below, the simple path is $[1,2,1]$. It is already a palindrome without any addition, so it is an addition palindrome sequence.

If you choose the leaf pair below, the simple path is $[1,2,1,2,1]$. It is already a palindrome, so it is an addition palindrome sequence.

If you choose the leaf pair below, the simple path is $[1,2,2]$. No sequence of operations turns it into a palindrome of length 2 or more, so it is not an addition palindrome sequence.

The first line contains the height $h$ of the tree. $(2\le h\le 10)$

The second line contains $A_1, A_2, \cdots, A_{2^h-1}$, separated by spaces. $(-1\,000\le A_i\le 1\,000)$

$A_i$ is the value written on node $i$.

The root is always node $1$, and the parent of node $i$ ($i\ge 2$) is $\lfloor i/2 \rfloor$.

All values in the input are integers.

On the first line, print the number of simple paths between pairs of leaf nodes that form addition palindrome sequences.