Enforcing relationships priority when using var-length matches


I am pretty new to Neo4j and Cypher. From my understanding, when using variable length pattern matching, Neo4j uses DFS by default.

I have the following query:
MATCH path = (u:Egg {name: "RootEgg"})-[:Mutation1|Mutation2*0..]->(c:Egg) RETURN c, relationships(path)

Starting from a node, I want to find all the child nodes (they form a binary tree) that are related with the root node by a series of mutations (either Mutation1 or Mutation2). The thing is I want this binary tree ordered such that when using DFS to traverse the tree, Mutation 1 has priority over Mutation 2.

Is there anyway to specify this relationship priority so DFS tries to expand Mutation1 before it tries to expand Mutation2.

Thank you very much!