<p>The ℓ-deck of a graph <i>G</i> is the multiset of all induced subgraphs of <i>G</i> on ℓ vertices. We say that a graph is reconstructible from its ℓ-deck if no other graph has the same ℓ-deck. In 1957, Kelly showed that every tree with <i>n</i> ≥ 3 vertices can be reconstructed from its (<i>n</i> − 1)-deck, and Giles strengthened this in 1976, proving that trees on at least 6 vertices can be reconstructed from their (<i>n</i> − 2)-decks. Our main theorem states that trees are reconstructible from their (<i>n</i> − <i>r</i>)-decks for all <i>r</i> ≤ <i>n</i>/9 + <i>o</i>(<i>n</i>), making substantial progress towards a conjecture of Nýdl from 1990. In addition, we can recognise the connectedness of a graph from its ℓ-deck when ℓ ≥ 9<i>n</i>/10, and reconstruct the degree sequence when <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\ell\ge\sqrt{2n\log(2n)}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mi>ℓ</mi> <mo>≥</mo> <msqrt> <mn>2</mn> <mi>n</mi> <mi>log</mi> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo stretchy="false">)</mo> </msqrt> </math></EquationSource> </InlineEquation>. All of these results are significant improvements on previous bounds.</p>