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We give an explicit construction of the scaling limit of the minimum spanningtree of the complete graph. The limit object is described using a recursiveconstruction involving the convex minorants of a Brownian motion with parabolicdrift (and countably many i.i.d. uniform random variables); we call it theBrownian parabolic tree.  Aside from the new representation, this point of view has multipleconsequences. For instance, it permits us to prove that its Hausdorff dimensionis almost surely 3. It also intrinsically contains information related to someunderlying dynamics: one notable by-product is the construction of a standardmetric multiplicative coalescent which couples the scaling limits of randomgraphs at different points of the critical window in terms of the same simplebuilding blocks.  The above results actually fit in a more general framework. They result fromthe introduction of a new family of continuum random trees associated withfunctions via their convex minorants, that we call convex minorant trees. Weinitiate the study of these structures in the case of Brownian-like paths. Inpassing, we prove that the convex minorant tree of a Brownian excursion is aBrownian continuum ranndom tree, and that it provides a coupling between theAldous--Pitman fragmentation of the Brownian continuum random tree and itsrepresentation by Bertoin.

Convex minorant trees associated with Brownian paths and the continuum limit of the minimum spanning tree