  We study structural properties of trees grown by preferential attachment. In
this mechanism, nodes are added sequentially and attached to existing nodes at
a rate that is strictly proportional to the degree. We classify nodes by their
depth n, defined as the distance from the root of the tree, and find that the
network is strongly stratified. Most notably, the distribution f_k^(n) of nodes
with degree k at depth n has a power-law tail, f_k^(n) ~ k^{-\gamma(n)}. The
exponent grows linearly with depth, gamma(n)=2+(n-1)/<n-1>, where the brackets
denote an average over all nodes. Therefore, nodes that are closer to the root
are better connected, and moreover, the degree distribution strongly varies
with depth. Similarly, the in-component size distribution has a power-law tail
and the characteristic exponent grows linearly with depth. Qualitatively, these
behaviors extend to a class of networks that grow by a redirection mechanism.
