  Combinatorial trees can be used to represent genealogies of asexual
individuals. These individuals can be endowed with birth and death times, to
obtain a so-called `chronological tree'. In this work, we are interested in the
continuum analogue of chronological trees in the setting of real trees. This
leads us to consider totally ordered and measured trees, abbreviated as TOM
trees. First, we define an adequate space of TOM trees and prove that under
some mild conditions, every compact TOM tree can be represented in a unique way
by a so-called contour function, which is right-continuous, admits limits from
the left and has non-negative jumps. The appropriate notion of contour function
is also studied in the case of locally compact TOM trees. Then we study the
splitting property of (measures on) TOM trees which extends the notion of
`splitting tree' studied in \cite{MR2599603}, where during her lifetime, each
individual gives birth at constant rate to independent and identically
distributed copies of herself. We prove that the contour function of a TOM tree
satisfying the splitting property is associated to a spectrally positive L\'evy
process that is not a subordinator, both in the critical and subcritical cases
of compact trees as well as in the supercritical case of locally compact trees.
