Reconstructing trees from small cards

The $\ell$-deck of a graph $G$ is the multiset of all induced subgraphs of $G$ on $\ell$ vertices. In 1976, Giles proved that any tree on $n\geq 6$ vertices can be reconstructed from its $\ell$-deck for $\ell \geq n-2$. Our main theorem states that it is enough to have $\ell\geq (8/9+o(1))n$, making substantial progress towards a conjecture of N\'ydl from 1990. In addition, we can recognise connectedness from the $\ell$-deck if $\ell\geq 9n/10$, and reconstruct the degree sequence from the $\ell$-deck if $\ell\ge \sqrt{2n\log(2n)}$. All of these results are significant improvements on previous bounds.Comment: 24 pages, fixed several typo