How to extract the sample split (values) of decision tree leaves ( terminal nodes) applying h2o library
So far I'm not seeing a way to extract training information from the model.  The H2OTree.predictions can/should give you proportion information, but won't give you leaf sample sizes.  For that, you should be able to use predict_leaf_node_assignment, passing your training set in (to wastefully get passed through the model, *shrug*).

predict_leaf_node_assignment should return a dataframe with the leaf assignment for each of your training points.  (The R version appears to support returning either the path or the node id, but the python one doesn't seem to have it.)  You could take this, join to the original frame, and use group and aggregation functions to produce the desired [n1,n2].*

Regarding the output of predictions, see  https://stackoverflow.com/questions/44735518/how-to-reproduce-the-h2o-gbm-class-probability-calculation .  In particular, the default learning rate in H2O's GBM is 0.1, which helps explain your muted results.

Finally, for a little more fun with the the model's tree objects, see https://www.pavel.cool/machine%20learning/h2o-3/h2o-3-tree-api/
and
https://novyden.blogspot.com/2018/12/finally-you-can-plot-h2o-decision-trees.html

*EDIT: For doing the grouping and aggregation:
(I'm more used to pandas than H2O frames, so I'll convert first.  And given that H2O thinks your FALSE class is the main class, maybe those are strings not boolean?)

predicted_leaves_frame = treemodel.predict_leaf_node_assignment(data).as_data_frame()
df = data.as_data_frame()
df['binary_dep_var'] = df['dep_var'].apply(lambda x: 1 if x=='TRUE' else 0)
df['T1'] = predicted_leaves_frame['T1.C1']
print(df.groupby('T1')['binary_dep_var'].agg(['sum','count','mean'])


This should give for each leaf the number of TRUE samples and the total number of samples and the ratio.  If you really need the number of FALSE samples, you could define your own aggregation function or just post-process this new dataframe.