Tree-Structured Trajectory Training

• Tree-Structured Trajectory Training is a set of methods that uses hierarchical tree models to represent sequential decision-making and capture both short- and long-term dependencies.
• These approaches decompose long-horizon trajectories into sub-goal trees, enabling efficient parallel computation and robust planning through dynamic programming.
• They integrate tree-based memory, generative sampling, and graph reasoning to support multimodal predictions in domains such as autonomous navigation and robotics.

Tree-structured trajectory training is a family of approaches in sequential decision modeling, trajectory prediction, and planning that represent data, latent memory, or trajectory hypotheses using hierarchical (tree-based) structures rather than traditional flat or linear forms. These methods capitalize on the recursive, branching nature of trees to capture short- and long-term dependencies, facilitate parallel computation, explicitly encode multimodality, and support more interpretable or efficient learning dynamics. The tree structure may be realized in memory architectures, via algorithmic decomposition (e.g., sub-goal trees), through generative sampling methods, or in the explicit representation of relational dependencies. Applications range from autonomous navigation and motion planning to structured code generation and general agent orchestration.

1. Tree-based Memory and Model Architectures

Tree-structured memory models, such as the Tree Memory Network (TMN) (Fernando et al., 2017), introduce hierarchical memory architectures for sequence modeling. TMN consists of three modules: an LSTM-based input encoder, a controller employing attention mechanisms, and a recursive memory modeled as a tree of S-LSTM cells. The tree structure allows bottom-up aggregation of hidden states, where lower levels encode short-term dependencies and higher levels capture abstract, long-term temporal patterns. Memory read operations utilize attention across tree nodes and fuse input and memory-derived embeddings for output prediction: $$y_t = \operatorname{ReLU}(W_\text{out} z_t + (1 - W_\text{out}) c_t)$$ where $z_t$ is the attended memory output and $c_t$ is the current input encoding.

The recursive update step aggregates child states with distinct forget and input gates: $$c_t^P = f_t^L \cdot c_{t-1}^L + f_t^R \cdot c_{t-1}^R + i_t \cdot \tanh(\beta)$$ with $\beta$ formed from affine combinations of child hidden states. This mechanism underpins abstraction and temporal generalization within the memory hierarchy, outperforming flat sequence models for long-horizon trajectory prediction tasks.

2. Hierarchical Trajectory Representation and Decomposition

Tree-structured trajectory training leverages hierarchical decompositions, as in sub-goal tree frameworks (Jurgenson et al., 2019), where a long-horizon trajectory from start $s$ to goal $g$ is recursively split into intermediate sub-goals. The joint probability for the trajectory is decomposed as: $$P(s_0, \ldots, s_T \mid s, g; \theta) = P(s_{T/2} \mid s, g; \theta) \cdot P(s_{T/4} \mid s, s_{T/2}; \theta) \cdot \ldots$$ Prediction can thus proceed in $O(\log T)$ parallel steps, improving computational efficiency and robustness to error propagation.

Optimization employs a dynamic programming equation tailored to sub-goal trees: $$V_k(s, s') = \min_{s_m} \left\{ V_{k-1}(s, s_m) + V_{k-1}(s_m, s') \right\}$$ This equation explicitly seeks the best intermediate splits and supports efficient planning across discrete or continuous state spaces. It provides empirical improvements over classical Bellman-based RL controllers in long-horizon, multi-modal planning domains.

3. Tree Structures for Multimodal Prediction and Interpretability

Tree-structured approaches are also employed to explicitly enumerate and optimize over multiple plausible future behaviors, supporting interpretability and multimodality. The Social Interpretable Tree (SIT) (Shi et al., 2022) for pedestrian prediction constructs a hand-crafted ternary tree, where branches correspond to maneuvers (“go straight,” “turn left,” “turn right”) at fixed intervals. A coarse-to-fine optimization strategy selects branches by attention scoring and refines predictions with teacher-forcing: $$p = \operatorname{Softmax}(\phi(F_s) \cdot \psi(F_\text{tree})^\top)$$ Subsequent losses combine cross-entropy, regression (Huber), and final refining losses: $$\mathcal{L} = \lambda_1 \mathcal{L}_\text{coarse} + \lambda_2 \mathcal{L}_\text{clf} + \lambda_3 \mathcal{L}_\text{ref}$$ The symbolic paths in the tree yield interpretable descriptions of predicted behaviors, aligning with requirements in safety-critical domains.

Goal-guided tree sampling, as in GDTS (Sun et al., 2023), merges generative diffusion models and tree-based trajectory refinement. A trunk stage employs deterministic DDPM steps to generate a common template, while branch stages refine diverse trajectories for sampled goals: $$Y^{k-1} = \frac{1}{\sqrt{\alpha_k}} \left(Y^k - \frac{1 - \alpha_k}{\sqrt{1 - \bar{\alpha}_k}} \epsilon(Y^k, k, f_*) \right)$$ This bifurcated process accelerates multi-modal inference and maintains diversity without the instability of GAN/VAE methods.

4. Tree Principles in Planning and Policy Synthesis

In decision-making tasks, tree-structured planning combines policy search with multimodal behavior models. Tree Policy Planning (TPP) (Chen et al., 2023) uses an ego trajectory tree to enumerate candidate motions, coupled with a scenario tree capturing multimodal agent forecasts (potentially ego-conditioned). Planning is cast as a discrete MDP: $J = \sum_{i=0}^N L_i(r^i, e^i)$ with dynamic programming performed across tree branches: $$Q(r^{i+1}, e^i) = L_i(r^i, e^i) + \sum_{e^{i+1} \in ch(e^i)} P(e^{i+1} | e^i) V(r^{i+1}, e^{i+1})$$ Optimal policies at each stage select the branch minimizing expected cost. This structure both improves safety and reduces conservativeness in complex, interactive environments.

5. Tree-based Aggregation, Learning Dynamics, and Bias Mitigation

Beyond model structure, tree principles inform training and post-processing. Trajectory Aggregation Tree (TAT) (Feng et al., 28 May 2024) is a training-free augmentation for diffusion planners: each trajectory is a branch, nodes represent aggregated state observations, and unreliable samples are pruned through weighted majority voting: $e^* = \arg\max_e Q(e), \quad Q(e) = E_V(e)$ TAT provably reduces the probability of artifact selection as the number of trajectories increases, as formalized in

$$p_\text{artifact}(n;\epsilon) < \frac{1}{2}\left[1 - \text{erf}\left(\frac{n/2 - n\epsilon}{\sqrt{2n\epsilon(1-\epsilon)}}\right)\right]$$

This aggregation enables effective mitigation of stochastic risks and accelerates planning.

For bias reduction, TrACT (Zhang et al., 18 Apr 2024) employs clustering of training samples by prediction error and variance, yielding difficulty “branches” (easy, trained, hard, confusing). Prototypical contrastive learning aligns embeddings with cluster leaders: $L_\text{protoNCE} = L_\text{ins} + L_\text{proto}$ with terms defined using densities and temperatures, and the overall objective combining regression and contrastive components: $L = L_\text{reg} + \lambda L_\text{protoNCE}$ This approach yields improved accuracy and scene compliance on long-tail data.

6. Structurally Enriched Trajectories and Graph-based Memory Systems

Tree structures can generalize into graph-based memory, as in Structurally Enriched Trajectories (SETs) and SETLE architecture (Catarau-Cotutiu et al., 17 Mar 2025). SETs extend trajectory representations beyond flat sequences, encoding hierarchical relations among objects, interactions, and affordances: $\mathcal{G}_\tau = (V, E, \Phi)$ where $V$ contains multi-type nodes, $E$ encodes temporal and structural edges, and $\Phi$ maps each node to high-dimensional embeddings. SETLE integrates SET episodes into a hierarchical, heterogeneous memory graph, using graph neural networks to encode subgraphs and leverage multi-episode relational knowledge for generalization. This paradigm supports cross-task transfer and robust compositional learning, especially in RL-generated data where agents must adapt to new dynamics and structural patterns.

7. Practical Implications, Limitations, and Applications

Tree-structured trajectory training methods have been demonstrated to outperform flat, sequence-based models in accuracy, robustness, and computational efficiency across domains such as autonomous driving, robotics, pedestrian motion prediction, and code generation. They enable efficient representation and learning for multimodal, long-horizon, and interpretable tasks, and can be flexibly integrated with both classical and deep learning frameworks. Scalability depends on managing memory complexity and tuning structural hyperparameters. Some approaches, such as TAT and SETLE, highlight the importance of aggregation and graph-based reasoning for stability, generalization, and lifelong learning.

Collectively, tree-structured trajectory training marks a substantial methodological advance, enabling richer contextual understanding, parallelization, and risk mitigation in sequential prediction, planning, and agent learning systems (Fernando et al., 2017, Jurgenson et al., 2019, Shi et al., 2022, Chen et al., 2023, Sun et al., 2023, Zhang et al., 18 Apr 2024, Feng et al., 28 May 2024, Catarau-Cotutiu et al., 17 Mar 2025).