Self-stabilization is a versatile approach to fault-tolerance since it permits a distributed system to recover from any transient fault that arbitrarily corrupts the contents of all memories in the system. Byzantine tolerance is an attractive feature of distributed systems that permits to cope with arbitrary malicious behaviors. We consider the well known problem of constructing a maximum metric tree in this context. Combining these two properties proves difficult: we demonstrate that it is impossible to contain the impact of Byzantine nodes in a self-stabilizing context for maximum metric tree construction (strict stabilization). We propose a weaker containment scheme called topology-aware strict stabilization, and present a protocol for computing maximum metric trees that is optimal for this scheme with respect to impossibility result.