If you do not add the $a|T|$ penalty term in your loss function, it will try to fit the training set as best as possible. This will render the model with only few options to "generalize" to perturbed/other data sets. Thus low insample bias will lead to higher test-set variance.
Remember, what you are really trying to do is not to fit the full complexity of a training set but to identify its "regularities" which are common to all data sets that are derived from the same underlying data generating process.
For that to happen, the model needs to avoid fitting noise, which will happen for overly complex models. Instead, restricted models will try to mimic lower dimensional (smooth) topological sub-spaces.
More formally, your training bias/optimisim can be defined as
$$
opt=Err_{in}-Err_{train}
$$
where $opt$ refers to the optimism in your training set. This is because $Err_{train}$ refers to your training error and $Err_{in}$ to your in-sample error, which arises from sampling from the same training distribution.
According to Tibshirani one can quite generally show that
$$
E[op]=\frac{2}{N}\sum_{i}^N Cov(\hat{y}_i,y_i)
$$
Thus, the amount by which your training error underestimates the true error depends on how strongly $y_i$ affects its own predictions $\hat{y}_i$. The harder we fit the data, the greater the $Cov$ term wil be, thereby increasing optimism.
Thus, the covariance will increase the more sophisticated and flexible your model is. For instance, in a classical regression model of the form $Y=f(X)+\epsilon$, the Covariance has a clear dependency with respect to the number of covariates/regressors $d$, ie,
$$
\sum_i Cov(\hat{y}_i,y_i)=d \sigma^2_\epsilon
$$
So your ultimate training cost function will look like
$$
E[Err_{in}=E[Err_{train}]+2\frac{d}{N}\sigma^2_\epsilon
$$
Equally, in decision trees pruning will ensure to limit the possibilities to (over-) fit your training data, although there is no functional relationship known how the prune level affects the training bias. However, the bias will be monotonously related to the tree complexity $|T|$, thus
$$
E[Err_{in}]\equiv E[Err_{train}]+ h(|T|)
$$
where $h$ is a monotonously increasing function. Thus, in the absence of further evidence, you assume linearity, i.e. $h(x)=a x$ (the constant term can be thrown away as it does not affect the optimization problem).