Two Mysterious Letters from a Tree

Your task is to extract $2$ letters from a binary tree which is composed of unsigned integers respecting the following rules. Let $n$ be the height of a tree. At the level $k$ (1 \le k \le n), the tree contains $k$ nodes and each node has $2$ children nodes (except the leaf nodes at the level $n$ which have no children). See the example below to understand the tree formation. Some nodes may have $2$ parent nodes.

Example:

You need to walk in a tree on the path that has a maximum summation (e.g. 1 + 5 + 9 = 15). Numbers in each summation cannot cross into different links (e.g. 5 + 7 is illegal). Then, your intermediate task is to calculate $2$ numbers for letter extraction. The first number is calculated from i^2 where $i$ is a number along the maximum summation path and $n$ is the height of a tree.

The second number is a summation of the maximum path ($i$). Regarding to the example above, the first number = 1 + 25 + 81 = $107$ and the second number = 1 + 5 + 9 = $15$.

Finally, these two numbers are transformed into two lower case letters from 'a' to 'z' respectively, where 'a' is used for $0$ and 'z' is used for $25$. Since there are only $26$ letters, a number greater than $25$ will reuse the same set of letters. For example $107$ = 'd' and $15$ = 'p' (that is, the first letter 'a' = $0$ or $26$ or $52$ etc).

Write a program to find the $2$ mysterious letters from a given tree.