To save even more data, replace every symbol with A. Then just store the amount of As.
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Would it be better to use A's and B's? We could half the necessary length of a symbol by assigning a certain combination of A's and B's to a symbol. And if characters take up too much space we could use 1's and 0's instead!
Then we could make a language where the letters are equivalent to the number of As
To save sending the tree every time, we could just have a fixed layout of letters and symbols. This would have the advantage that we could put them in order, which would be easier to work with.
Before I opened this article I thought "I bet they just learned what Huffman Encoding is" and I stand my hypothesis. It's an extremely fun compression algorithm as it's very trivial to comprehend and reason about... it's straightforward enough you can write it yourself!