Let tw(G) denote the treewidth of graph G. Given a graph G and a positive integer k such that tw(G) <= k + 1, we are to decide if tw(G) <= k. We give a certifying algorithm RTW ("R" for recursive) for this task: it returns one or more tree-decompositions of G of width <= k if the answer is YES and a minimal contraction H of G such that tw(H) > k otherwise. RTW uses a heuristic variant of Tamaki's PID algorithm for treewidth (ESA2017), which we call HPID. RTW, given G and k, interleaves the execution of HPID with recursive calls on G /e for edges e of G, where G / e denotes the graph obtained from G by contracting edge e. If we find that tw(G / e) > k, then we have tw(G) > k with the same certificate. If we find that tw(G / e) <= k, we "uncontract" the bags of the certifying tree-decompositions of G / e into bags of G and feed them to HPID to help progress. If the question is not resolved after the recursive calls are made for all edges, we finish HPID in an exhaustive mode. If it turns out that tw(G) > k, then G is a certificate for tw(G') > k for every G' of which G is a contraction, because we have found tw(G / e) <= k for every edge e of G. This final round of HPID guarantees the correctness of the algorithm, while its practical efficiency derives from our methods of "uncontracting" bags of tree-decompositions of G / e to useful bags of G, as well as of exploiting those bags in HPID. Experiments show that our algorithm drastically extends the scope of practically solvable instances. In particular, when applied to the 100 instances in the PACE 2017 bonus set, the number of instances solved by our implementation on a typical laptop, with the timeout of 100, 1000, and 10000 seconds per instance, are 72, 92, and 98 respectively, while these numbers are 11, 38, and 68 for Tamaki's PID solver and 65, 82, and 85 for his new solver (SEA 2022).