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Nous donnons des r\'esultats d'\'equidistribution d'\'el\'ements de corps defonctions sur des corps finis, et d'irrationnels quadratiques sur ces corps,dans leurs corps locaux compl\'et\'es. Nous d\'eduisons ces r\'esultats deth\'eor\`emes d'\'equidistribution de perpendiculaires communes dans desquotients d'arbres par des r\'eseaux de leur groupe d'automorphismes,d\'emontr\'es \`a l'aide de propri\'et\'es ergodiques du flot g\'eod\'esiquediscret.  Non-Archimedean equidistribution and group actions on trees. We giveequidistribution results of elements of function fields over finite fields, andof quadratic irrationals over these fields, in their completed local fields. Wededuce these results from equidistribution theorems of common perpendiculars inquotients of trees by lattices in their automorphism groups, proved by usingergodic properties of the discrete geodesic flow.

Equidistribution non-archim\'edienne et actions de groupes sur les arbres