  In the Kingman coalescent tree the length of order $r$ is defined as the sum
of the lengths of all branches that support $r$ leaves. For $r=1$ these
branches are external, while for $r\ge2$ they are internal and carry a subtree
with $r$ leaves. In this paper we prove that for any $s\in\mathbb{N}$ the
vector of rescaled lengths of orders $1\le r\le s$ converges to the
multivariate standard normal distribution as the number of leaves of the
Kingman coalescent tends to infinity. To this end we use a coupling argument
which shows that for any $r\ge2$ the (internal) length of order $r$ behaves
asymptotically in the same way as the length of order 1 (i.e., the external
length).
