Since order within each relation matters, the following is a different rule:

{{x, y}} -> {{z, y}, {y, x}}

This rule can be represented graphically as:

RulePlot[ResourceFunction[
"WolframModel"][{{x, y}} -> {{z, y}, {y, x}}],
VertexLabels -> Automatic, "RulePartsAspectRatio" -> 0.25]

Like the previous rule, running this rule also gives a tree, but now with a somewhat different structure:

ResourceFunction[
"WolframModel"][{{{x, y}} -> {{z, y}, {y, x}}}, {{1, 2}},
8]["StatesPlotsList", "MaxImageSize" -> 180]

With the other rendering from above, the last 3 steps here are:

Graph[Rule @@@ #, GraphLayout -> "LayeredDigraphEmbedding",
VertexStyle ->
ResourceFunction["WolframPhysicsProjectStyleData"]["SpatialGraph",
"VertexStyle"],
EdgeStyle ->
ResourceFunction["WolframPhysicsProjectStyleData"]["SpatialGraph",
"EdgeLineStyle"], AspectRatio -> 1/3] & /@
Take[ResourceFunction[
"WolframModel"][{{{x, y}} -> {{z, y}, {y, x}}}, {{1, 2}}, 8,
"StatesList"], -3]