A long-standing question of Stanley asks whether non-isomorphic trees can have the same chromatic symmetric function (CSF). To attack this question, we analyze the generalized degree polynomial (GDP), a graph invariant introduced by Crew that enumerates subsets of vertices by size and number of internal and boundary edges. Aliste-Prieto et al. showed that for trees, the GDP is linearly determined by the CSF, making it a useful intermediary to study Stanley's question. We present several new classes of data about a tree that can be recovered from the GDP, such as the double-degree sequence, which enumerates pairs of adjacent vertices by degree, and the leaf adjacency sequence, which enumerates vertices by degree and number of adjacent leaves. Then, by considering a variant of the GDP for polarized trees -- trees with distinguished "left" and "right" vertices -- we prove a recurrence relation for the GDP, and present methods to construct arbitrarily large sets of trees with the same GDP.