  As a generalization of random recursive trees and preferential attachment
trees, we consider random recursive metric spaces. These spaces are constructed
from random blocks, each a metric space equipped with a probability measure,
containing a labelled point called a hook, and assigned a weight. Random
recursive metric spaces are equipped with a probability measure made up of a
weighted sum of the probability measures assigned to its constituent blocks. At
each step in the growth of a random recursive metric space, a point called a
latch is chosen at random according to the equipped probability measure and a
new block is chosen at random and attached to the space by joining together the
latch and the hook of the block. We use martingale theory to prove a law of
large numbers and a central limit theorem for the insertion depth; the distance
from the master hook to the latch chosen. We also apply our results to further
generalizations of random trees, hooking networks, and continuous spaces
constructed from line segments.
