Editorial for Pondo Euler Tour

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Approach

Root the tree at vertex $1$ ~1~ and run a depth-first search. Append a vertex to the tour when the search enters it, recurse on its neighbours in increasing order (skipping the parent), then append it again when the search leaves. Every vertex is written twice, so the tour has length $2n$ ~2n~.

Building the adjacency lists and running the search both take $O(n)$ ~O(n)~ time after sorting neighbours, which is $O(n \log n)$ ~O(n \log n)~ in the worst case (a star).

Solution (Python)

import sys

sys.setrecursionlimit(1_000_000)

data = list(map(int, sys.stdin.buffer.read().split()))
n = data[0]
adj = [[] for _ in range(n)]
idx = 1
for _ in range(n - 1):
    u = data[idx] - 1
    v = data[idx + 1] - 1
    idx += 2
    adj[u].append(v)
    adj[v].append(u)

for u in range(n):
    adj[u].sort()

tour = []


def dfs(u, p):
    tour.append(u + 1)
    for v in adj[u]:
        if v != p:
            dfs(v, u)
    tour.append(u + 1)


dfs(0, -1)
print(*tour)

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