ABSTRACT We use the Millennium and Millennium-II simulations to illustrate the Tessellation-Level-Tree (tlt), a hierarchical tree structure linking density peaks in a field constructed by voronoi tessellation of the particles in a cosmological N-body simulation. The tlt uniquely partitions the simulation particles into disjoint subsets, each associated with a local density peak. Each peak is a subpeak of a unique higher peak. The tlt can be persistence filtered to suppress peaks produced by discreteness noise. Thresholding a peak’s particle list at $∼ 80\left 〈 ρ \right 〉 \,$ results in a structure similar to a standard friend-of-friends halo and its subhaloes. For thresholds below $∼ 7\left 〈 ρ \right 〉 \,$, the largest structure percolates and is much more massive than other objects. It may be considered as defining the cosmic web. For a threshold of $5\left 〈 ρ \right 〉 \,$, it contains about half of all cosmic mass and occupies $∼ 1{{\ \rm per\ cent}}$ of all cosmic volume; a typical external point is then ∼7h−1 Mpc from the web. We investigate the internal structure and clustering of tlt peaks. Defining the saddle point density ρlim as the density at which a peak joins its parent peak, we show the median value of ρlim for FoF-like peaks to be similar to the density threshold at percolation. Assembly bias as a function of ρlim is stronger than for any known internal halo property. For peaks of group mass and below, the lowest quintile in ρlim has b ≈ 0, and is thus uncorrelated with the mass distribution.