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By studying an admissible family of branching mechanisms introduced in Li(2014), we obtain a pruning procedure on L\'evy trees. Then we could constructa decreasing L\'evy-CRT-valued process $\{{\mathcal T}_t\}$ by pruning L\'evytrees and an analogous process $\{{\mathcal T}^*_t\}$ by pruning a criticalL\'evy tree conditioned to be infinite. Under a regular condition on theadmissible family of branching mechanisms, we show that the law of $\{{\mathcalT}_t\}$ at the ascension time can be represented by $\{{\mathcal T}^*_t\}$. Theresults generalize those studied in Abraham and Delmas (2012).

A tree-valued Markov processes associated with an admissible family of branching mechanisms