When you fit a Decision Tree, observations in a leaf have the same predicted value. If there are n leaves, then the Actual by Predicted plot shows at most n distinct predicted values. This gives the plot the appearance of having points arranged on vertical lines. Each of these lines corresponds to a predicted value for some leaf.
Actual by Predicted Plots for Boston Housing Data
The ROC curve is for categorical responses. The classical definition of ROC curve involves the count of True Positives by False Positives as you accumulate the frequencies across a rank ordering. The True Positive y-axis is labeled “Sensitivity” and the False Positive X-axis is labeled “1-Specificity”. If you slide across the rank ordered predictor and classify everything to the left as positive and to the right as negative, this traces the trade-off across the predictor's values.
ROC curves are nothing more than a curve of the sorting efficiency of the model. The model rank-orders the fitted probabilities for a given Y-value. Starting at the lower left corner, the curve is drawn up when the row comes from that category and to the right when the Y is another category.
In the following picture, the Y axis shows the number of Ys where Y=1, and the X axis shows the number of Ys where Y=0.
ROC for Perfect Fit
Because partitions contain clumps of rows with the same (that is tied) predicted rates, the curve actually goes slanted, rather than purely up or down.
Lift Curve