  Learning representations for query plans play a pivotal role in machine
learning-based query optimizers of database management systems. To this end,
particular model architectures are proposed in the literature to transform the
tree-structured query plans into representations with formats learnable by
downstream machine learning models. However, existing research rarely compares
and analyzes the query plan representation capabilities of these tree models
and their direct impact on the performance of the overall optimizer. To address
this problem, we perform a comparative study to explore the effect of using
different state-of-the-art tree models on the optimizer's cost estimation and
plan selection performance in relatively complex workloads. Additionally, we
explore the possibility of using graph neural networks (GNNs) in the query plan
representation task. We propose a novel tree model BiGG employing Bidirectional
GNN aggregated by Gated recurrent units (GRUs) and demonstrate experimentally
that BiGG provides significant improvements to cost estimation tasks and
relatively excellent plan selection performance compared to the
state-of-the-art tree models.
