An Analysis of Random d -Dimensional Quad Trees
Luc Devroye, Louise Laforest · SIAM Journal on Computing · 1990
It is shown that the depth of the last node inserted in a random quad tree constructed from independent uniform $[0,1]^d $ random vectors is in probability asymptotic to $({2 / d}) \log n$, where log denotes the natural logarithm. In addition, for $d = 2$, exact values are obtained for all the moments of the depth of the last node.