  Splitting trees are those random trees where individuals give birth at
constant rate during a lifetime with general distribution, to i.i.d. copies of
themselves. The width process of a splitting tree is then a binary, homogeneous
Crump--Mode--Jagers (CMJ) process, and is not Markovian unless the lifetime
distribution is exponential. Here, we allow the birth rate to be infinite, that
is, pairs of birth times and lifespans of newborns form a Poisson point process
along the lifetime of their mother, with possibly infinite intensity measure. A
splitting tree is a random (so-called) chronological tree. Each element of a
chronological tree is a (so-called) existence point $(v,\tau)$ of some
individual $v$ (vertex) in a discrete tree, where $\tau$ is a nonnegative real
number called chronological level (time). We introduce a total order on
existence points, called linear order, and a mapping $\varphi$ from the tree
into the real line which preserves this order. The inverse of $\varphi$ is
called the exploration process, and the projection of this inverse on
chronological levels the contour process. For splitting trees truncated up to
level $\tau$, we prove that thus defined contour process is a L\'evy process
reflected below $\tau$ and killed upon hitting 0. This allows to derive
properties of (i) splitting trees: conceptual proof of Le Gall--Le Jan's
theorem in the finite variation case, exceptional points, coalescent point
process, age distribution; (ii) CMJ processes: one-dimensional marginals,
conditionings, limit theorems, asymptotic numbers of individuals with infinite
vs finite descendances.
