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Let $M_d$ be the moduli space of one-dimensional complex polynomial dynamicalsystems. The escape rates of the critical points determine a critical heightsmap $G: M_d \to \mathbb{R}^{d-1}$. For generic values of $G$, each connectedcomponent of a fiber of $G$ is the deformation space for twist deformations onthe basin of infinity. We analyze the quotient space $\mathcal{T}_d^*$ obtainedby collapsing each connected component of a fiber of $G$ to a point. The space$\mathcal{T}_d^*$ is a parameter-space analog of the polynomial tree $T(f)$associated to a polynomial $f:\mathbb{C}\to\mathbb{C}$, studied by DeMarco andMcMullen, and there is a natural projection from $\mathcal{T}_d^*$ to the spaceof trees $\mathcal{T}_d$. We show that the projectivization$\mathbb{P}\mathcal{T}_d^*$ is compact and contractible; further, the shiftlocus in $\mathbb{P}\mathcal{T}_d^*$ has a canonical locally finite simplicialstructure. The top-dimensional simplices are in one-to-one corespondence withtopological conjugacy classes of structurally stable polynomials in the shiftlocus.

Critical heights on the moduli space of polynomials