Why is the high-pass filter result in a discrete wavelet transform (DWT) downsampled?
Assume the original sampling rate is $F_s$ and that the filters are perfect brickwall Halfband filters. After the low pass filtering the frequency content is from $0 \rightarrow \frac{F_s}{4}$. Because of the new bandwidth, the sampling rate can be reduced to $F_s/2$.

For the high-pass filter, the frequency content lies in the range $\frac{F_s}{4} \rightarrow \frac{F_s}{2}$. There is no content in the range $0 \rightarrow \frac{F_s}{4}$. We can downsample by a factor of 2 in this case as well. The downsampling causes the frequency content in the $\frac{F_s}{4} \rightarrow \frac{F_s}{2}$ to be aliased into the range $0 \rightarrow \frac{F_s}{4}$, but because there is no frequency content in this range, due to the high-pass filter, we have not lost any information.