  Huffman coding is a widely used method for lossless data compression because
it optimally stores data based on how often the characters occur in Huffman
trees. An $n$-ary Huffman tree is a connected, cycle-lacking graph where each
vertex can have either $n$ "children" vertices connecting to it, or 0 children.
Vertices with 0 children are called \textit{leaves}. We let $h_n(q)$ represent
the total number of $n$-ary Huffman trees with $q$ leaves. In this paper, we
use a recursive method to generate upper and lower bounds on $h_n(q)$ and get
$h_2(q) \approx (0.1418532)(1.7941471)^q+(0.0612410)(1.2795491)^q$ for $n=2$.
This matches the best results achieved by Elsholtz, Heuberger, and Prodinger in
August 2011. Our approach reveals patterns in Huffman trees that we used in our
analysis of the Binary-Ternary (BT) trees we created. Our research opens a
completely new door in data compression by extending the study of Huffman trees
to BT trees. Our study of BT trees paves the way for designing data-specific
trees, minimizing possible wasted storage space from Huffman coding. We prove a
recursive formula for the number of BT trees with $q$ leaves. Furthermore, we
provide analysis and further proofs to reach numeric bounds. Our discoveries
have broad applications in computer data compression. These results also
improve graphical representations of protein sequences that facilitate in-depth
genome analysis used in researching evolutionary patterns.
