Huffman Tree Depth, Is there any theory?
I don't know of a way to compute the length of the code exactly without constructing a Huffman code. And there may be more than one optimal Huffman code for a given set of weighted items, with different lengths. But there has been some related theory on lengths of Huffman codes:

Length-limited Huffman coding is a variant of Huffman coding where you are not allowed to use codewords longer than a given threshold. One can construct an optimal length-limited code in time $O(nL)$ where $n$ is the number of items and $L$ is the length bound; it is unknown whether this can be improved to match the time of unrestricted Huffman coding. See Larmore and Hirschberg, "A fast algorithm for optimal length-limited Huffman codes", JACM 1990, and Turpin and Moffat, "Practical length-limited coding for large alphabets", Computer J. 1995.

The length-limited problem was introduced earlier by Gilbert, "Codes based on inaccurate source probabilities", IEEE Trans. Inf. Th. 1971, who also showed that it is possible to restrict the length to log_2 n + (log log n)^{1+o(1)} while increasing the expected length by at most one. In "Restructuring ordered binary trees", J. Alg. 2004, Evans and Kirkpatrick write that restricting the length more strongly, to ceiling(log_2 n)+1, can force the expected length increase to be arbitrarily close to one. Evans and Kirkpatrick show that this strong length restriction can be obtained, even for alphabetic codes, with an expected increase in length of two.