# Reconstructing Trees from Traces

## TL;DR

This paper introduces algorithms for reconstructing node-labeled trees from deletion channel traces, achieving polynomial trace complexity for certain tree classes, contrasting with the exponential complexity in string trace reconstruction.

## Contribution

It presents the first polynomial-trace algorithms for reconstructing specific classes of trees, extending trace reconstruction beyond strings.

## Key findings

- Polynomial number of traces suffices for complete trees and spiders
- Contrasts with exponential trace requirements in string trace reconstruction
- Uses novel combinatorial and complex analytic techniques

## Abstract

We study the problem of learning a node-labeled tree given independent traces from an appropriately defined deletion channel. This problem, tree trace reconstruction, generalizes string trace reconstruction, which corresponds to the tree being a path. For many classes of trees, including complete trees and spiders, we provide algorithms that reconstruct the labels using only a polynomial number of traces. This exhibits a stark contrast to known results on string trace reconstruction, which require exponentially many traces, and where a central open problem is to determine whether a polynomial number of traces suffice. Our techniques combine novel combinatorial and complex analytic methods.