How do you find the height of the recurrence tree $T(n,k)=T(\frac{n}{2},k)+T(n,\frac{k}{4})+nk$
Here is a recursive definition of a height of a tree $T$:

If $T$ is a leaf, then its height is 0 (some people prefer to use 1).
Otherwise, the height of $T$ is the maximum height of any of its children, plus 1.

Equivalently, the height of a tree is the maximum length of a root-to-leaf path (in edges; if you measure it in vertices, leaves should have height 1).
This makes it clear that the correct recurrence for the height is
$$
H(n,k) = \max(H(\tfrac{n}{2},k), H(n,\tfrac{k}{4})) + 1,
$$
with an appropriate base case. The solution is roughly $\max(\log_2 n, \log_4 k)$, depending on the base case.