Folder.java

import javax.xml.bind.annotation.XmlAccessType;
import javax.xml.bind.annotation.XmlAccessorType;
import javax.xml.bind.annotation.XmlAttribute;
import javax.xml.bind.annotation.XmlElement;
import javax.xml.bind.annotation.XmlRootElement;
import java.util.ArrayList;
import java.util.stream.Stream;


@XmlAccessorType(XmlAccessType.FIELD)
@XmlRootElement(name = "folder")
public class Folder {
    @XmlAttribute(name = "name")
    private String name;

    @XmlElement(name = "folder")
    private ArrayList<Folder> folderList;

    public String getName() {
        return name;
    }


    public ArrayList<Folder> getFoldersList() {
        return folderList;
    }


    public Stream<Folder> extractTree() {
        return folderList == null ?
                Stream.of(this) :
                Stream.concat(
                    Stream.of(this),
                    folderList.stream().flatMap(Folder::extractTree)
                );
    }


}

Main.java

import javax.xml.bind.JAXBContext;
import javax.xml.bind.JAXBException;
import javax.xml.bind.Unmarshaller;
import java.io.ByteArrayInputStream;
import java.io.UnsupportedEncodingException;
import java.nio.charset.StandardCharsets;
import java.util.Collection;
import java.util.stream.Collectors;

public class Main {


    public static Collection<String> folderNames(String xml, char startingLetter) {
        try {
            JAXBContext jaxbContext = JAXBContext.newInstance(Folder.class);
            Unmarshaller jaxbUnmarshaller = jaxbContext.createUnmarshaller();

            Folder folder = (Folder) jaxbUnmarshaller.unmarshal(
                    new ByteArrayInputStream(
                            xml.getBytes(StandardCharsets.UTF_8.name())
                    )
            );

            return folder.extractTree()
                    .map(Folder::getName)
                    .filter(s -> s.startsWith(String.valueOf(startingLetter)))
                    .collect(Collectors.toList());

        } catch (JAXBException | UnsupportedEncodingException e) {
            throw new RuntimeException(e.getMessage());
        }
    }

    public static void main(String[] args) throws Exception {
        String xml =
                "<?xml version=\"1.0\" encoding=\"UTF-8\"?>" +
                        "<folder name=\"c\">" +
                        "<folder name=\"program files\">" +
                        "<folder name=\"uninstall information\"><folder name=\"unified folder\" /></folder>" +
                        "</folder>" +
                        "<folder name=\"users\" />" +
                        "</folder>";

        Collection<String> names = folderNames(xml, 'u');
        names.forEach(System.out::println);
    }


}